Problem 72 - Applied Calculus

Question

Find the indefinite integrals. $$\int \cos \theta d \theta$$

Step-by-Step Solution

Answer

\( \sin \theta + C \)

Step 1: Recall the Basic Integral Formula

To solve the integral \( \int \cos \theta \, d\theta \), it's important to recall the formula for the integral of the cosine function. The integral of \( \cos \theta \) with respect to \( \theta \) is \( \sin \theta \), since the derivative of \( \sin \theta \) is \( \cos \theta \).

Step 2: Apply the Integral Formula

Using the information from Step 1, apply the formula such that \( \int \cos \theta \, d\theta = \sin \theta + C \), where \( C \) is the constant of integration.

Key Concepts

Integration Formulas

Integration formulas are essential tools in calculus used to find the antiderivative or the indefinite integral of functions. An indefinite integral is represented by the symbol \( \int \) and is followed by a function and the variable of integration, such as \( \int f(x) \, dx \). The result of this process is a family of functions, each differing by a constant. This constant of integration, often denoted as \( C \), accounts for the fact that many functions have the same derivative.

For many common functions, integration formulas can directly provide these antiderivatives. Knowing these formulas helps streamline the integration process significantly, allowing us to bypass more complex methods such as substitution or integration by parts for simpler functions.

• Example: One of the most basic integration formulas is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), assuming \( n eq -1 \).
• Another example pertinent to trigonometric functions is \( \int \cos x \, dx = \sin x + C \).

These formulas serve as building blocks, making it easier to tackle more complex calculus problems.

Trigonometric Integrals

Trigonometric integrals involve integrals of trigonometric functions like sine, cosine, tangent, and others. Knowing how to integrate these functions is crucial because they frequently appear in different areas of mathematics and physics.

The process often relies on memorized integration formulas for basic trigonometric functions. For instance, the integral of \( \cos \theta \) is \( \sin \theta + C \). This comes from the derivative rule that tells us the derivative of \( \sin \theta \) is \( \cos \theta \).

• Similarly, \( \int \sin x \, dx = -\cos x + C \), because the derivative of \(-\cos x\) is \( \sin x \).
• For \( \int \tan x \, dx \), the result is \(-\ln|\cos x| + C\).

Utilizing these integral formulas for trigonometric functions simplifies problems, allowing quick and efficient solutions without extensive calculation.

Basic Integrals

Basic integrals refer to the standard antiderivatives of fundamental functions, making them essential for solving integrals quickly and efficiently. These basic integrations serve as a foundation for more intricate integrals.

In the realm of basic integrals, knowing a set of primary rules is paramount. One of the simplest is understanding that \( \int c \, dx = cx + C \), where \( c \) is a constant. This reflects how constants are embedded in the integration process.

• For example, \( \int x^n \, dx \) as mentioned earlier is fundamental, especially for polynomials.
• Constant multiples lead to \( \int a f(x) \, dx = a \int f(x) \, dx \).

These basic integrals are fundamental tools that, when combined with other formulas or techniques, enable the integration of more complex functions seamlessly. Understanding them enriches one's ability to analyze and compute integrals efficiently.